Math 270B: Numerical Analysis (Part B) - Bo Li

Math 270B: Numerical Analysis (Part B)
Winter quarter 2023

Description This is the second part of a series of three parts of an introductory numerical analysis course for graduate students. The entire course series serves as a graduate qualifying exam course. The second part focuses on nonlinear equations, approximation, interpolation, and numerical integration. (More detailed topics are listed below on this web page.)
Prerequisites  Math 270A or consent of instructor.
Lectures 10:00 - 10:50, Mondays, Wednesdays, and Fridays, AP&M 5402.
Instructor Professor Bo Li
Office: AP&M 5723. Office phone: (858) 534-6932. E-mail:
Office hours: 11:00 - 12:00, Mondays and Fridays.
Teaching Assistant  Mr. Zunding Huang
Office: HSS 3084. E-mail:
Office hours: 10:00 am - 12:00 noon, Thursdays.
Textbook A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, 2nd ed., Springer, 2007. (UCSD e-version available.) (Some additional references are listed below on this web page.)
Lecture Notes Lecture notes for approximation, interpolation, and numerical integration will be posted in the course Canvas, and referred as Notes here.
Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7 - 9 (Notes: Sec. 3.1, 3.2, and Supplement), Lecture 10 - 13 (Notes: Sec. 3.3 - 3.6), Lecture 14 (Midterm exam), Lecture 15 - 26 (Notes: Sec. 3.7, 3.8, 4.1 - 4.6, 5.1 - 5.6).
Homework Assigned, collected, and graded regularly. We will use Gradescope.
Exams There will be one midterm exam and one final, close-book and close-note exams. The final exam will be cumulative.
        Midterm exam: 10:00 am - 10:50 am, Friday, February 10, AP&M 5402.
        Final exam: 10:30 am - 12:30 pm, Friday, March 24. (Place to be announced later.)
Note: Neither make-up nor rescheduled exams will be allowed unless a proved written excuse (such as hospitalization, a family emergency, and a major religious activity) is provided sufficiently ahead of time.
Grading The final course grade will be determined based on the homework and exams with the weight: homework - 30%, midterm exam 20%, and final exam - 50%.
Disability Accommodations Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter (paper or electronic) issued by the Office for Students with Disabilities (OSD) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments.
Academic Integrity All students are expected to conduct themselves with academic integrity. Violations of academic integrity will be treated seriously. See UCSD Academic Integrity page

Topics to be Covered
  1. Nonlinear Equations and Optimization (~ 3 lectures)
    1. Introduction
    2. Newton iteration
    3. Fixed-point iteration
  2. Optimization (~ 3 lectures)
    1. The gradient descent method
    2. The method of Lagrange multipliers
    3. The penalty method
  3. Polynomial Approximation (~ 8 lectures)
    1. The Weierstrass Theorem
    2. Best uniform approximations
    3. Least-squares approximations
    4. Orthogonal polynomials
  4. Polynomial Interpolation (~ 8 lectures)
    1. Lagrange interpolation
    2. Newton's formula and divided differences
    3. Hermite interpolation
    4. Piecewise polynomial interpolation
  5. Numerical Integration (~ 5 lectures)
    1. Interpolatory quadrature
    2. Euler-Maclaurin formula
    3. Gaussian quadrature
  1. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, 1966.
  2. E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 2nd ed., Wiley, 2010.
  3. P. J. Davis, Interpolation and Approximation, Dover, 1975.
  4. P. J. Davis and P. H. Rabinowitz, Methods of Numerical Integration, Academic Press, 1975.
  5. C. de Boor, A Practical Guide to Splines, Springer-Verlag, 2001.
  6. H. Engles, Numerical Quadrature and Cubature, Academic Press, 1980.
  7. R. Fletcher, Practical Methods of Optimization, 2nd ed., Wiley, 1987.
  8. O. Guler, Foundations of Optimization, Springer, 2010. (UCSD e-version available.)
  9. F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed., Dover, 1987.
  10. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, 1994.
  11. G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, 1966.
  12. J. M. Ortega and W. C. Rheinboldt, Iterarive Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
  13. R. Plato, Concise Numerical Mathematics, Amer. Math. Soc., 2003.
  14. P. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.
  15. T. J. Rivlin, Introduction to the Approximation of Functions, Dover, 1987.
  16. A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2006.
  17. I. Schoenberg, Cardinal Spline Functions, SIAM, 1973.
  18. L. L. Schumaker, Spline Functions: Basic Theory, 3rd ed., Cambridge University Press, 2007.
  19. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer, 2004. (UCSD e-version available.)
  20. A. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, 1971.
  21. E. Suli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, 2003.
  22. G. Szgo, Orthogonal Polynomials, 3rd ed., Amer. Math. Soc., 1967.

Last updated by Bo Li on March 14, 2023.